|Kac's program in kinetic theory|
|Time：||Wed/Fri 13:00–14:50, 2014-10-8~2014-10-31|
|Instructor：||Kleber Carrapatoso [Fondation Mathématique Jacques Hadamard]|
|Place：||Conference Room 1, Floor 1, Jin Chun Yuan West Building|
We shall present in these lectures some recent developments on the Kac’s program in kinetic theory, in particular for the Boltzmann and Landau equations.
In his seminal work on the 50’s, Kac introduced the notion of “chaos” and “propagation of chaos” in order to rigorously derive kinetic equations from particle models in the limit where the number of particles goes to infinity, and also to relate properties of the solutions of the kinetic equations to properties of the corresponding particle system, this is now known as Kac’s program.
In the first part of the lectures we will discuss about the chaos property of a family of N-particle probability distributions, as well as stronger notions of chaos using the entropy and Fisher information functionals.
In the second part, we will present results concerning the propagation of chaos for the spatially homogeneous Boltzmann and Landau equations. More precisely we will obtain estimates of propagation of chaos that are quantitative in the number of particles and uniform in time. Moreover we will prove a relaxation to equilibrium for the many-particle system that is uniform in the number of particles.
Notions on partial differential equations, probability and functional analysis.
- Kac, M. Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III (Berkeley and Los Angeles, 1956), University of California Press, pp. 171–197.
- Carrapatoso, K. Quantitative and qualitative Kac’s chaos on the Boltzmann’s sphere. Preprint arXiv:1205.1241, to appear in Ann. Inst. H. Poincaré Probab. Statist.
- Carrapatoso, K., and Einav, A. Chaos and entropic chaos in Kac's model without high moments. Electron. J. Probab. 18 (2013), no. 78, 1-38.
- Mischler, S., and Mouhot, C. Kac’s program in kinetic theory. Inv. Math. 193, 1 (2013), 1–147.